broadening

Electron-impact broadening models. The default GBK/Ferri model evaluates a component half-width using a density-temperature prefactor, shell radius, and a dynamical cutoff factor,

\[\gamma_e(\Delta\omega) = W_\mathrm{pref}\,\langle r^2\rangle_n \left[C_n + G(\Delta\omega)\right],\]

with

\[G(\Delta\omega) = \frac{1}{2}E_1(y), \qquad y = \left(\frac{n^2}{2Z}\right)^2 \frac{\Delta\omega^2 + \omega_c^2}{\mathrm{Ry}\,T_e}.\]

The cutoff is chosen as

\[\omega_c = \max\left(\omega_p,\omega_e,\omega_L,\omega_{\alpha\alpha'}\right).\]

Electron impact broadening operator and collision G-functions (Lee, GBK, and Dufty RPA).

starkzee.broadening.calculate_plasma_frequency(Ne_m3)[source]

Return the electron plasma angular frequency ω_p [rad s⁻¹].

The plasma frequency sets the lower cutoff for electron-impact broadening: perturbations with frequency ω < ω_p are screened by collective plasma oscillations and do not contribute to individual collisions.

ω_p = √(N_e e² / (ε₀ m_e))

Parameters:: Ne_m3 (float) – Electron number density [m⁻³].
Returns:: Plasma angular frequency [rad s⁻¹].
Return type:: float

starkzee.broadening.calculate_larmor_frequency(B)[source]

Return the electron Larmor (cyclotron) angular frequency ω_L [rad s⁻¹].

In a magnetic field B the electron gyrates at

ω_L = e B / m_e

This frequency acts as a lower cutoff for the GBK electron-broadening model when it exceeds both ω_p and ω_e: cyclotron motion prevents an electron from approaching the radiator more closely than the cyclotron radius, reducing the effective cross-section at low detunings.

Parameters:: B (float) – Magnetic field strength [T]. B=0 returns 0 safely.
Returns:: Larmor angular frequency [rad s⁻¹].
Return type:: float

starkzee.broadening.calculate_configuration_frequency(Ne_m3, Te_ev)[source]

Return the configuration-change frequency ω_e = 1 / τ_e [rad s⁻¹].

τ_e is the mean time for an electron to cross the Wigner-Seitz radius r_e at the thermal velocity v_th:

r_e = (3 / 4π N_e)^(1/3) — Wigner-Seitz inter-electron radius v_th = √(k_B T_e / m_e) — thermal speed τ_e = r_e / v_th ω_e = 1 / τ_e = v_th / r_e

Note on the 2π factor. Ferri, Peyrusse & Calisti (2022) write ω_e = 2π / τ_e, which is dimensionally correct: 1/τ_e is a rate in Hz and multiplying by 2π gives an angular frequency in rad/s comparable to ω_p and ω_L. However, that formula yields ω_e ≈ 290 meV at Ne = 10¹⁷ cm⁻³, Te = 5 eV, which exceeds ω_L even at B = 1 kT (ω_L ≈ 116 meV) and renders the max() cutoff insensitive to B. Their published figure shows distinct curves at B = 500 T and 1 kT, consistent only with ω_e < ω_L(500 T) ≈ 58 meV. Their code (PPPB) therefore likely used 1/τ_e directly as an angular frequency (a common plasma-physics convention conflating collision rate and angular frequency), giving ω_e ≈ 46 meV. StarkZee adopts the same convention.

Parameters:

Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].

Returns:

Configuration-change frequency ω_e = v_th / r_e [rad s⁻¹].

Return type:

float

starkzee.broadening.calculate_debye_length(Ne_m3, Te_ev)[source]

Return the classical electron Debye screening length λ_D [m].

λ_D = √(ε₀ k_B T_e / (e² N_e))

Parameters:

Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].

Returns:

Debye length [m].

Return type:

float

starkzee.broadening.calculate_cutoff_kappa_m(Z, n, Te_ev)[source]

Return the maximum wave-number cutoff κ_m [m⁻¹] (ZEST convention).

κ_m = min(κ_geo, κ_thermal) where:: κ_geo = Z / (n² a₀) — geometric (Bohr orbit) κ_thermal = Z √(2 m_e k_B T_e) / (ħ n²) — thermal de Broglie

The geometric limit dominates at T_e > 13.6 eV; the thermal limit dominates below that threshold (1 Rydberg).

Parameters:

Z (int) – Nuclear charge.
n (int) – Principal quantum number of the upper level.
Te_ev (float) – Electron temperature [eV].

Returns:

Maximum wave-number cutoff κ_m [m⁻¹].

Return type:

float

starkzee.broadening.calculate_electron_impact_prefactor(Ne_m3, Te_ev)[source]

Return the electron-impact width prefactor W₀ [eV].

The total electron-impact half-width is

W_e = W₀ × ⟨r²⟩_n × [C_n + G(Δω)]

where ⟨r²⟩_n is the statistically averaged squared radius of the upper level (in a₀²), C_n is a strong-collision constant, and G is the GBK dynamical factor. W₀ combines the electron density, temperature, and fundamental constants into a single prefactor:

W₀ = (4π/3) N_e √(2m_e / (π k_B T_e)) × (ħ/m_e)² × (ħ/e)

The factor (ħ/m_e)² converts the squared velocity integral from SI to atomic-unit area, and (ħ/e) converts rad s⁻¹ to eV.

Parameters:

Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].

Returns:

Prefactor W₀ [eV a₀⁻²]. Multiply by ⟨r²⟩_n [a₀²] to obtain [eV].

Return type:

float

starkzee.broadening.gbk_model(delta_omega_ev, omega_c_ev, Te_ev, Z, n=2)[source]

Evaluate the semi-classical GBK dynamical broadening function G(Δω).

The Griem–Baranger–Kolb (GBK) model accounts for the frequency dependence of electron-impact broadening using an exponential-integral form:

G(Δω) = ½ E₁(y)

where E₁ is the exponential integral and the dimensionless argument is

y = (n² / 2Z)² × (Δω² + ω_c²) / (E_H T_e)

with E_H = e²/2a₀ = 13.6057 eV (Rydberg energy) and T_e in eV. GBK, Ferri and ZEST all use this same definition. At line center (Δω = 0) G reduces to ½ E₁((n²/2Z)² ω_c² / (E_H T_e)) ≈ a positive constant; in the far wings where Δω ≫ ω_c the argument y grows and G → 0, suppressing the broadening at large detunings (the impact approximation breaks down).

The cutoff frequency ω_c = max(ω_p, ω_e, ω_L, ω_αα′) prevents the logarithm from diverging at small impact parameters and encodes the transition from the impact regime to the quasi-static regime.

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning from line center Δω [eV].
omega_c_ev (float) – Cutoff angular frequency ω_c [eV] (= ħ ω_c in SI units).
Te_ev (float) – Electron temperature [eV].
Z (int) – Nuclear charge of the radiating ion.
n (int, optional) – Principal quantum number of the upper level (default 2).

Returns:

Dimensionless GBK factor G(Δω) ≥ 0.

Return type:

float or ndarray

References

Griem, Kolb & Shen, Phys. Rev. 116, 4 (1959) — original G-function for hydrogen. Griem, Baranger, Kolb & Oertel, Phys. Rev. 125, 177 (1962) — helium extension. Ferri, Peyrusse & Calisti, Matter Radiat. Extremes 7, 015901 (2022) — C_n constants and ω_c formulation.

starkzee.broadening.gbk_zest_model(delta_omega_ev, Ne_m3, Te_ev, Z, n=2)[source]

Evaluate the GBK G-function G(Δω) with ω_p as the only cutoff (ZEST convention).

Uses the κ_m-based argument written in eV units:

x = κ_m λ_D (dimensionless cutoff) arg = (Δω² + ω_p²) / (2 x² ω_p²) G = ½ E₁(arg)

where Δω and ω_p are in eV, and κ_m = min(Z/(n²a₀), Z√(2mₑkBTe)/(ħn²)).

This is equivalent to the analytic form (n²/2Z)²(Δω²+ωp²)/(Ryd×Te) only in the geometric κ_m regime (Te > 13.6 eV). Below that threshold the thermal branch of κ_m is active and the analytic form gives a different (larger) G. This implementation uses the physical κ_m so it is consistent with lee_model() and dufty_model(): all three converge at large Δω where G → ½ E₁(Δω²/(2x²ωp²)).

The only cutoff frequency is ω_p (no ω_e, no ω_L).

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning Δω [eV].
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
Z (int) – Nuclear charge.
n (int, optional) – Principal quantum number of the upper level (default 2).

Returns:

Dimensionless GBK factor G(Δω) ≥ 0.

Return type:

float or ndarray

See also

gbk_model: GBK/Ferri formula using max(ω_p, ω_e, ω_L) as cutoff.
lee_model: Lee analytical blend of impact and wing limits.

starkzee.broadening.lee_model(delta_omega_ev, Ne_m3, Te_ev, Z, n=2)[source]

Evaluate Lee’s analytical approximation for the G-function G(Δω).

The Lee model blends two limiting expressions:: G₀ = ½ [ln(1 + x²) − x²/(1 + x²)], x = κ_m λ_D (impact limit, Δω → 0) G∞ = ½ E₁(Δω² / (2 κ_m² v_th²)) (wing limit, Δω → ∞) G = min(G₀, G∞)

Unlike gbk_zest_model(), the G∞ argument contains only Δω² (not Δω² + ω_p²): the plasma cutoff is captured implicitly by G₀.

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning Δω [eV].
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
Z (int) – Nuclear charge.
n (int, optional) – Principal quantum number of the upper level (default 2).

Returns:

Lee G-function value G(Δω) ≥ 0.

Return type:

float or ndarray

starkzee.broadening.dufty_model(delta_omega_ev, Ne_m3, Te_ev, Z, n=2)[source]

Evaluate the Dufty RPA G-function G(Δω) by numerical integration.

Integrates the electron RPA dielectric function over wave numbers κ_min to κ_m, accounting for Landau damping via the Dawson function:

G(Δω) = ∫_{κ_min}^{κ_m} e^{−x²} / (κ ‖ε(κ, Δω)‖²) dκ

where x = abs(Δω) / (κ v_th) and v_th = √(2 k_B T_e / m_e). This is the most accurate of the three G models but also the slowest (one quadrature per point).

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning Δω [eV].
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
Z (int) – Nuclear charge.
n (int, optional) – Principal quantum number of the upper level (default 2).

Returns:

Dufty RPA G-function value G(Δω) ≥ 0.

Return type:

float or ndarray

See also

gbk_zest_model: Faster closed-form GBK approximation.

starkzee.broadening.electron_impact_width(delta_omega_ev, Ne_m3, Te_ev, B, Z, n=2)[source]

Return the total electron-impact half-width W_e(Δω) [eV].

Implements the frequency-dependent GBK model for electron Stark broadening, extended to include a magnetic-field-dependent cutoff frequency. The total half-width (HWHM of the Lorentzian) is:

W_e(Δω) = W₀ × ⟨r²⟩_n × [C_n + G(Δω, ω_c)]

Prefactor W₀ — see calculate_electron_impact_prefactor().

Mean squared radius ⟨r²⟩_n is the statistical (2l+1)-weighted average of ⟨r²⟩_{n,l} over all l subshells:

⟨r²⟩_{n,l} = (n²/2Z²) [5n² + 1 − 3l(l+1)] [a₀²] ⟨r²⟩_n = (1/n²) Σ_{l=0}^{n-1} (2l+1) ⟨r²⟩_{n,l}

Scales approximately as n⁴/Z², so broadening grows rapidly with n.

Strong-collision constant C_n (Ferri, Peyrusse & Calisti, Matter Radiat. Extremes 7, 015901 (2022), Table 1):

n	C_n
≤ 2	1.50
3	1.00
4	0.75
5	0.50
> 5	0.40

Cutoff frequency ω_c = max(ω_p, ω_L, ω_e, ω_αα′) (Ferri, Peyrusse & Calisti 2022, below Eq. 20), where ω_p is the plasma frequency, ω_L the electron Larmor frequency, ω_e = 1/τ_e the configuration-change frequency (see calculate_configuration_frequency()), and ω_αα′ the state-to-state transition frequency. ω_αα′ = 0 for hydrogen (degenerate l-subshells within a shell); it is non-zero for multi-electron atoms and is currently a zero placeholder pending implementation.

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning from line center Δω [eV]. Pass 0.0 for the on-resonance (line-center) width.
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
B (float) – Magnetic field [T]. B=0 is valid; ω_L = 0 in that case.
Z (int) – Nuclear charge of the radiating ion (1 for hydrogen).
n (int, optional) – Principal quantum number of the upper level (default 2). The width refers to the upper-level broadening only; the lower-level contribution is neglected, consistent with the semi-classical model.

Returns:

Electron-impact half-width W_e [eV] (HWHM of the Lorentzian component at detuning Δω). Always positive.

Return type:

float or ndarray

Notes

A negligible floor of 1e-10 eV is added by the caller in calculate_static_profile() solely to prevent 0/0 in the Lorentzian at exactly zero density; the raw value returned here is the physical width without that floor.

starkzee.broadening.electron_impact_width_zest(delta_omega_ev, Ne_m3, Te_ev, Z, n=2, model='gbk')[source]

Return the electron-impact half-width W_e(Δω) [eV] using the ZEST broadening model.

Implements the ZEST electron broadening operator:

W_e(Δω) = W₀ × ⟨r²_intra⟩_n × [G_n + G(Δω)]

Prefactor W₀ = (4π/3) N_e √(2m_e/πkT_e) (ħ/m_e)² (ħ/e).

Intra-shell squared radius ⟨r²_intra⟩_n is the mean over the n²-degenerate spatial basis of the intra-shell dipole sum:

r²_intra,i = Σ_{j same shell} ‖⟨i|r|j⟩‖²

which uses only within-shell matrix elements, unlike the full ⟨r²⟩_{n,l} used by electron_impact_width(). For hydrogen-like ions the per-l value is

r²_intra,l = (9n²/4Z²) (n² − l(l+1) − 1) [a₀²]

derived from the intra-shell radial element ⟨n,l|r|n,l±1⟩ = (3n/2Z)√(n²−(l±1)²) and the angular sum factors C(l, l+1) = (l+1)/(2l+1), C(l, l−1) = l/(2l+1). The shell average is exact in closed form:

⟨r²_intra⟩_n = (9n²/8Z²)(n²−1) [a₀²]

This matches the average computed in ZEST’s _get_level_widths_sp.

The difference from electron_impact_width() is in both the G-function and r²:

r²: intra-shell sum ⟨r²_intra⟩ < full diagonal ⟨r²⟩ (ratio ~0.41 at n=2, ~0.53 at n=3)
Minimum impact parameter: ZEST uses ρ_min = 1/κ_m (temperature-dependent); StarkZee uses the fixed n²a₀/(2Z).
Cutoff frequency: ZEST uses ω_p only; StarkZee uses max(ω_p, ω_e, ω_L).
G_n / C_n values: identical in both.

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning Δω [eV].
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
Z (int) – Nuclear charge.
n (int, optional) – Principal quantum number of the upper level (default 2).
model ({'gbk', 'lee', 'dufty'}, optional) – G-function approximation: 'gbk' → gbk_zest_model(), 'lee' → lee_model(), 'dufty' → dufty_model().

Returns:

Electron-impact half-width W_e [eV].

Return type:

float or ndarray

See also

electron_impact_width: StarkZee/Ferri GBK with fixed ρ_min and ω_e, ω_L cutoffs.

starkzee.broadening.electron_impact_width_model(delta_omega_ev, Ne_m3, Te_ev, B, Z, n=2, electron_model='pppb')[source]

Return the electron-impact half-width W_e(Δω) [eV] from the selected model.

Thin dispatcher that lets the profile solvers switch between the two published electron-impact prescriptions without changing any call site:

'pppb' (default) → electron_impact_width(). The PPPB / Ferri, Peyrusse & Calisti (2022) form: fixed minimum impact parameter ρ_min = n²a₀/(2Z), GBK G-function, and a B-dependent cutoff ω_c = max(ω_p, ω_e, ω_L).
'zest' / 'zest-gbk' → electron_impact_width_zest() (model='gbk').
'zest-lee' → ZEST with Lee’s analytic G-function (model='lee').
'zest-dufty' → ZEST with the Dufty RPA G-function (model='dufty').

The ZEST variants use ω_p as the only cutoff (κ_m-based ρ_min, no Larmor term), consistent with the ZEST formulation, so B does not enter them; it is accepted here only to give every model a uniform signature.

Parameters:

delta_omega_ev (float or array-like) – Frequency detuning from line center Δω [eV].
Ne_m3 (float) – Electron number density [m⁻³].
Te_ev (float) – Electron temperature [eV].
B (float) – Magnetic field [T]. Used only by the 'pppb' model (Larmor cutoff).
Z (int) – Nuclear charge.
n (int, optional) – Principal quantum number of the upper level (default 2).
electron_model (str, optional) – One of 'pppb', 'zest', 'zest-gbk', 'zest-lee', 'zest-dufty' (default 'pppb').

Returns:

Electron-impact half-width W_e [eV].

Return type:

float or ndarray

See also

electron_impact_width: PPPB model.
electron_impact_width_zest: ZEST model (GBK / Lee / Dufty G-functions).

starkzee.broadening.electron_impact_r2_scaling(eigenvectors, n, Z)[source]

Per-eigenstate ⟨k|r²|k⟩ / ⟨r²⟩_avg — the electron-impact operator diagonal.

The semi-classical electron-impact width is linear in the upper-state operator R⃗·R⃗ = r² (Ferri/PPPB Eq. 19; ZEST Eq. 8). StarkZee’s scalar width functions evaluate it with the shell-averaged ⟨r²⟩_avg; ZEST and PPPB instead use the operator’s value resolved on each Stark-Zeeman dressed state. Because the width is linear in r², the operator-diagonal width of dressed state k is simply

W_e^{op}(k) = W_e^{scalar} × ⟨k|r²|k⟩ / ⟨r²⟩_avg ,

i.e. the scalar width times the factor returned here.

r² is purely radial, so in the |n, l, m_l, m_s⟩ basis it is diagonal with value ⟨r²⟩_{n,l} = (n²/2Z²)[5n²+1−3l(l+1)]; the Stark-Zeeman eigenstates mix l, so ⟨k|r²|k⟩ varies from state to state. The factor averages to 1 over the shell (trace preserved), redistributing width among components.

This is the diagonal of the broadening operator. Keeping the diagonal only (and discarding off-diagonal ⟨k|r²|k’⟩) is exactly the c_k = 0 approximation that recovers the ZEST operator. The off-diagonal part is what generates the complex SDT intensity a_k + i c_k of the full PPPB operator (non-Hermitian Liouvillian) — not yet implemented; see the REVIEW note below.

Parameters:

eigenvectors (ndarray, shape (2n², 2n²)) – Stark-Zeeman eigenvectors (columns) in the |n,l,m_l,m_s⟩ basis, as returned by the profile solvers’ eigh call.
n (int) – Principal quantum number of the shell these eigenvectors belong to.
Z (int) – Nuclear charge.

Returns:

Per-eigenstate scaling factor ⟨k|r²|k⟩ / ⟨r²⟩_avg (mean ≈ 1).

Return type:

ndarray, shape (2n²,)

Notes

REVIEW (future): only the operator diagonal is used (c_k = 0 → ZEST). The full PPPB operator keeps the off-diagonal ⟨k|r²|k’⟩, builds the non-Hermitian Liouvillian, and yields the complex intensity a_k + i c_k (estimated ~1–2 % asymmetry for Hβ at 1 kT). Implementing it also calls for the lower-manifold d†·d piece (currently neglected, as in the scalar model). See FFM_implementation_plan.md.